Radiation relativity question????

Question

A pulsar is receding directly away from the Earth with a speed of 8.0 C, a starship is sent towards the pulsar with a speed of 0.95 C relative to the earth. If an earth observer sees 153 pulses emitted every second, what rate does the starship see the pulses?

CORRECTION!!!! PULSAR IS RECEDING AT 0.8C

Answer 1

I regret to state that Captain Mephisto is incorrect about the speed of the pulsar: under conditions of continuing acceleration, it is possible to "outrun" a photon, but not while going at constant speed - however great.

In the rest frame of the Earth, the pulse frequency is 153 Hz.

As seen in that frame, at one instant a pulse blasts past the nose of the rocket; another pulse is just distance L = c/153 beyond it. The second pulse and the rocket are coincident at the time = L/(c + 0.95*c) = (L/c)*(1/1.95); at that time, the rocket will have moved 0.95*c *(L/c)*(1/1.95) = (0.95/1.95)L = (0.95/1.95)*(c/153).

So in the Earth frame:
dt = (L/c)*(1/1.95)
dx = (0.95/1.95)*L

In the starship frame, the coordinates are x' and t'. Because of the invariance of the interval (minkowski metric),
c^2*dt'^2 - dx'^2 = c^2*dt^2 - dx^2
= L^2/(1.95)^2 - L^2*(0.95/1.95)^2
= L^2 * (1 - (0.95)^2)/(1.95^2)
= 0.0256*L^2

In the starship frame, dx' = 0, because both pulses were seen at the same spot, so
dt'^2 = 0.0256*(L/c)^2 = 0.0256/(153^2) = 1.095e-6, so:
dt' = sqrt(1.095e-6) = 1.0466e-3

Therefore, the rate of pulses seen by the starship is:
1/dt' = 1/1.0466e-3 = 955.5 Hz

Answer 2

Sorry, a pulsar can NOT move with a speed of 8c relative to the earth. If the space between earth and the pulsar is expanding at such a rate then no light from the pulsar will be seen. In this case the distance between us and the pulsar will be increasing at 8c but the pulsar itself can be at rest with respect to space or traveling at some speed less than the speed of light. But the pulsar contains mass so it can never even reach the speed of light.